Technical Field
This disclosure relates to integrated circuits (ICs). More specifically, this disclosure relates to determining the resistance of a conducting structure.
Related Art
Advances in process technology and an almost insatiable appetite for consumer electronics have fueled a rapid increase in the size and complexity of IC designs. Accurate estimation of parasitic resistances has become very important for accurately performing subsequent timing or signal integrity analyses. Since miniaturization is expected to continue relentlessly, accurate estimation of parasitic resistances is expected to become even more critical in the future.
A number of techniques can be used to determine the resistance of a conducting structure. These techniques include, but are not limited to, Finite Element Method (FEM), Boundary Element Method (BEM), and Fast Multipole Method (FMM). All of these techniques create a matrix equation (which represents a system of linear equations), and determine the resistance by solving the matrix equation. A significant amount of computational time and computational resources are spent on constructing and solving the matrix equation.
FEM techniques discretize the entire area of the conducting structure (or discretize the entire volume if the conducting structure is a 3-D structure). In contrast, BEM techniques discretize only the boundaries of the conducting structure. Therefore, BEM techniques typically result in a smaller matrix than FEM techniques. In BEM, an integral equation is formulated based on the linear partial differential equations that describe the physical phenomenon of interest, and BEM attempts to find a solution based on the constraint that a given set of boundary conditions must be satisfied. BEM techniques are well known in the art. For example, details of BEM can be found in numerous references, such as Becker, A. A. The Boundary Element Method in Engineering: A Complete Course. London: McGraw-Hill, 1992, which is herein incorporated by reference to provide details of BEM.
FIGS. 1A-1C illustrate how BEM can be used to determine resistance of a conducting structure. Boundary 102 of a conducting structure can be discretized to obtain a set of boundary elements 104, which are shown in FIG. 1A using dashed lines. As shown in FIG. 1B, a value (e.g., current density, voltage, etc.) associated with a given boundary element, e.g., boundary element 106, can be computed at the mid-point 108 of the boundary element 106. In a typical BEM approach, two values—(1) a potential, and (2) a partial derivative of the potential along a normal direction to the boundary element—are associated with each boundary element. At each boundary element, one of the two values is known, and the other value is unknown. The values associated with each boundary element are linearly related to other values associated with other boundary elements, and this linear relationship can be represented using a matrix equation.
In FIG. 1C, matrix equation 110 includes coefficient matrix 112, unknown vector 114, and constant vector 116. If the number of boundary elements is N, then there are a total of 2N values in the BEM problem (because there are two values associated with each boundary element). Of these 2N values, N are known and the other N are unknown. The values in constant vector 116 are constants which are calculated based on the boundary conditions at the boundary elements. Unknown vector 114 contains the variables whose values are to be determined. Once matrix equation 110 has been created, it can be solved to determine unknown vector 114, thereby determining the N unknown values.
When BEM is used to determine the resistance, the two values associated with each boundary element can be (1) the electric potential (i.e., voltage), and (2) the partial derivative of the electric potential along the normal direction to the boundary element. Note that the partial derivative of the electrical potential is proportional to the current density, so the two values associated with each boundary element can be (1) the voltage at the mid-point of the boundary element, and (2) the current density at the mid-point of the boundary element.
The constant vector 116 can include values that are computed based on known voltages and known current densities at the boundary elements. The coefficients in coefficient matrix 112 are based on the relationships between the voltages and current densities of the boundary elements, and can be computed based on the physical properties (e.g., resistivity, shape, dimensions, etc.) of the conducting structure. Once matrix equation 110 has been constructed, it can be solved to determine the unknown vector 114, thereby obtaining voltages and current densities at all boundary elements. These voltage and current density values can then be used to determine the resistance of the conducting structure.
FMM techniques can be used to improve the computational efficiency for generating coefficient matrix 112 (see e.g., Liu, Y. j., and N. Nishimura. “The Fast Multipole Boundary Element Method for Potential Problems: A Tutorial.” Engineering Analysis with Boundary Elements 30.5 (2006): 371-81). However, the size of the matrix can still be very large and it can take an unacceptably large amount of computational time and resources to construct the matrix and to determine the resistance by using existing BEM or FMM techniques. Hence, what are needed are apparatuses and techniques for accurately and efficiently determining the resistance of a conducting structure.